\(\int \frac {1+e^{-1+e^5} (-2+e^2 x^4)}{-e^2 x^4+2 e^{1+e^5} x^4} \, dx\) [7429]

Optimal result

Integrand size = 43, antiderivative size = 30 \[ \int \frac {1+e^{-1+e^5} \left (-2+e^2 x^4\right )}{-e^2 x^4+2 e^{1+e^5} x^4} \, dx=\frac {1}{3 e^2 x^3}+\frac {3+x}{2-e^{1-e^5}} \]

[Out]

(3+x)/(2-exp(1-exp(5)))+1/3*exp(-1)^2/x^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {6, 12, 14} \[ \int \frac {1+e^{-1+e^5} \left (-2+e^2 x^4\right )}{-e^2 x^4+2 e^{1+e^5} x^4} \, dx=\frac {1}{3 e^2 x^3}-\frac {e^{e^5} x}{e-2 e^{e^5}} \]

[In]

Int[(1 + E^(-1 + E^5)*(-2 + E^2*x^4))/(-(E^2*x^4) + 2*E^(1 + E^5)*x^4),x]

[Out]

1/(3*E^2*x^3) - (E^E^5*x)/(E - 2*E^E^5)

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1+e^{-1+e^5} \left (-2+e^2 x^4\right )}{\left (-e^2+2 e^{1+e^5}\right ) x^4} \, dx \\ & = -\frac {\int \frac {1+e^{-1+e^5} \left (-2+e^2 x^4\right )}{x^4} \, dx}{e^2-2 e^{1+e^5}} \\ & = -\frac {\int \left (e^{1+e^5}+\frac {e-2 e^{e^5}}{e x^4}\right ) \, dx}{e^2-2 e^{1+e^5}} \\ & = \frac {1}{3 e^2 x^3}-\frac {e^{e^5} x}{e-2 e^{e^5}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {1+e^{-1+e^5} \left (-2+e^2 x^4\right )}{-e^2 x^4+2 e^{1+e^5} x^4} \, dx=-\frac {\frac {-e+2 e^{e^5}}{3 x^3}+e^{2+e^5} x}{e^2 \left (e-2 e^{e^5}\right )} \]

[In]

Integrate[(1 + E^(-1 + E^5)*(-2 + E^2*x^4))/(-(E^2*x^4) + 2*E^(1 + E^5)*x^4),x]

[Out]

-(((-E + 2*E^E^5)/(3*x^3) + E^(2 + E^5)*x)/(E^2*(E - 2*E^E^5)))

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27

method	result	size
norman	\(\frac {\left (\frac {{\mathrm e} \,{\mathrm e}^{{\mathrm e}^{5}} {\mathrm e}^{-1} x^{4}}{2 \,{\mathrm e}^{{\mathrm e}^{5}} {\mathrm e}^{-1}-1}+\frac {{\mathrm e}^{-1}}{3}\right ) {\mathrm e}^{-1}}{x^{3}}\)	\(38\)
default	\(\frac {{\mathrm e}^{-2} \left (x \,{\mathrm e}^{{\mathrm e}^{5}+1}-\frac {-2 \,{\mathrm e}^{{\mathrm e}^{5}-1}+1}{3 x^{3}}\right )}{2 \,{\mathrm e}^{{\mathrm e}^{5}-1}-1}\)	\(39\)
gosper	\(\frac {\left (3 x^{4} {\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{5}-1}+2 \,{\mathrm e}^{{\mathrm e}^{5}-1}-1\right ) {\mathrm e}^{-2}}{3 x^{3} \left (2 \,{\mathrm e}^{{\mathrm e}^{5}-1}-1\right )}\)	\(44\)
parallelrisch	\(\frac {\left (3 x^{4} {\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{5}-1}+2 \,{\mathrm e}^{{\mathrm e}^{5}-1}-1\right ) {\mathrm e}^{-2}}{3 x^{3} \left (2 \,{\mathrm e}^{{\mathrm e}^{5}-1}-1\right )}\)	\(44\)
risch	\(\frac {6 x^{4} {\mathrm e}^{2 \,{\mathrm e}^{5}-2}-3 x^{4} {\mathrm e}^{{\mathrm e}^{5}-1}+4 \,{\mathrm e}^{-4+2 \,{\mathrm e}^{5}}-4 \,{\mathrm e}^{{\mathrm e}^{5}-3}+{\mathrm e}^{-2}}{3 \left (2 \,{\mathrm e}^{{\mathrm e}^{5}-1}-1\right )^{2} x^{3}}\)	\(58\)

[In]

int(((x^4*exp(1)^2-2)*exp(exp(5)-1)+1)/(2*x^4*exp(1)^2*exp(exp(5)-1)-x^4*exp(1)^2),x,method=_RETURNVERBOSE)

[Out]

(exp(1)/(2*exp(exp(5))*exp(-1)-1)*exp(exp(5))*exp(-1)*x^4+1/3/exp(1))/x^3/exp(1)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {1+e^{-1+e^5} \left (-2+e^2 x^4\right )}{-e^2 x^4+2 e^{1+e^5} x^4} \, dx=-\frac {{\left (3 \, x^{4} e^{2} + 2\right )} e^{\left (e^{5} + 1\right )} - e^{2}}{3 \, {\left (x^{3} e^{4} - 2 \, x^{3} e^{\left (e^{5} + 3\right )}\right )}} \]

[In]

integrate(((x^4*exp(1)^2-2)*exp(exp(5)-1)+1)/(2*x^4*exp(1)^2*exp(exp(5)-1)-x^4*exp(1)^2),x, algorithm="fricas"
)

[Out]

-1/3*((3*x^4*e^2 + 2)*e^(e^5 + 1) - e^2)/(x^3*e^4 - 2*x^3*e^(e^5 + 3))

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {1+e^{-1+e^5} \left (-2+e^2 x^4\right )}{-e^2 x^4+2 e^{1+e^5} x^4} \, dx=\frac {- x e^{2} e^{e^{5}} - \frac {- e + 2 e^{e^{5}}}{3 x^{3}}}{- 2 e^{2} e^{e^{5}} + e^{3}} \]

[In]

integrate(((x**4*exp(1)**2-2)*exp(exp(5)-1)+1)/(2*x**4*exp(1)**2*exp(exp(5)-1)-x**4*exp(1)**2),x)

[Out]

(-x*exp(2)*exp(exp(5)) - (-E + 2*exp(exp(5)))/(3*x**3))/(-2*exp(2)*exp(exp(5)) + exp(3))

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {1+e^{-1+e^5} \left (-2+e^2 x^4\right )}{-e^2 x^4+2 e^{1+e^5} x^4} \, dx=-\frac {x e^{\left (e^{5}\right )}}{e - 2 \, e^{\left (e^{5}\right )}} + \frac {e^{\left (-2\right )}}{3 \, x^{3}} \]

[In]

integrate(((x^4*exp(1)^2-2)*exp(exp(5)-1)+1)/(2*x^4*exp(1)^2*exp(exp(5)-1)-x^4*exp(1)^2),x, algorithm="maxima"
)

[Out]

-x*e^(e^5)/(e - 2*e^(e^5)) + 1/3*e^(-2)/x^3

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57 \[ \int \frac {1+e^{-1+e^5} \left (-2+e^2 x^4\right )}{-e^2 x^4+2 e^{1+e^5} x^4} \, dx=-\frac {x e^{\left (e^{5} + 1\right )}}{e^{2} - 2 \, e^{\left (e^{5} + 1\right )}} - \frac {2 \, e^{\left (e^{5} - 1\right )} - 1}{3 \, x^{3} {\left (e^{2} - 2 \, e^{\left (e^{5} + 1\right )}\right )}} \]

[In]

integrate(((x^4*exp(1)^2-2)*exp(exp(5)-1)+1)/(2*x^4*exp(1)^2*exp(exp(5)-1)-x^4*exp(1)^2),x, algorithm="giac")

[Out]

-x*e^(e^5 + 1)/(e^2 - 2*e^(e^5 + 1)) - 1/3*(2*e^(e^5 - 1) - 1)/(x^3*(e^2 - 2*e^(e^5 + 1)))

Mupad [B] (verification not implemented)

Time = 12.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93 \[ \int \frac {1+e^{-1+e^5} \left (-2+e^2 x^4\right )}{-e^2 x^4+2 e^{1+e^5} x^4} \, dx=\frac {{\mathrm {e}}^{-2}}{3\,x^3}+\frac {x\,{\mathrm {e}}^2}{2\,{\mathrm {e}}^2-{\mathrm {e}}^{3-{\mathrm {e}}^5}} \]

[In]

int(-(exp(exp(5) - 1)*(x^4*exp(2) - 2) + 1)/(x^4*exp(2) - 2*x^4*exp(exp(5) - 1)*exp(2)),x)

[Out]

exp(-2)/(3*x^3) + (x*exp(2))/(2*exp(2) - exp(3 - exp(5)))